Abstract

We prove optimal embeddings of the generalized Sobolev spaces , where is a rearrangement invariant function space, into the generalized Hölder-Zygmund space generated by a function space .

1. Introduction

The classical Sobolev space , , consists of all locally integrable functions , defined on , , with the Lebesgue measure, such that the following norm is finite: , where stands for the -norm. In investigating the regularity of the function , we may assume, without any loss of generality, that , is a domain in , and is zero outside . For simplicity we suppose that the Lebesgue measure of equals one and that the origin lies in . It is well known that in the supercritical case ,
where , is the Hölder-Zygmund space (see [1]). In the critical case the function may not be even continuous. The result (1) is not optimal. We prove that the optimal one is obtained if in (1) is replaced by the Marcinkiewicz space . In this paper we prove similar optimal results, when is replaced by a more general rearrangement invariant space . The Sobolev space consists of all with a finite quasinorm . More precisely, we consider quasinormed rearrangement invariant spaces , consisting of functions , such that the quasinorm , where is a monotone quasinorm, defined on with values in and is the cone of all locally integrable functions on with the Lebesgue measure. Monotonicity means that implies . We suppose that , which means continuous embeddings. Here is the decreasing rearrangement of , given by , and is the distribution function of , defined by denoting Lebesgue -measure. Note that for . Finally, .

Let , be the Boyd indices of . For example, if , then and the condition means . Note that for this is always satisfied. For these reasons we suppose that for the general , and the case is called super-critical, while the case -critical. In the super-critical case the function is always continuous, while the spaces in the critical case , can be divided into two subclasses: in the first subclass the functions may not be continuous—then the target space is rearrangement invariant, while these functions in the second subclass are continuous and the target space is the generalized Hölder-Zygmund space (see Definition 1). The separating space for these two subclasses is given by the Lorentz space , . If ; then consists of continuous functions (see the classical result of Stein [2]).

The main goal of this paper is to prove optimal embeddings of the Sobolev space into the generalized Hölder-Zygmund space . First we prove that this embedding for is equivalent to the continuity of the operator . The case is reduced to the continuity of by using the lifting principle ([1]). Moreover, if, for example, , then in the super-critical case, we can replace by the operator of multiplication . This implies a very simple characterization of both optimal target space and optimal domain space . Namely, the quasinorm in the optimal target space is given by and the quasinorm in the optimal domain space is given by . Note that we do not require to be rearrangement invariant. In the critical case, the formula for the optimal target space is more complicated. In some cases it can be simplified. To this end, we apply the -method of extrapolation ([3]) from the super-critical case. As a byproduct, we also characterize the embedding , where consists of all functions with bounded and uniformly continuous derivatives up to order . Namely, this is equivalent to the embedding if . The embedding is always true since .

The problem of the optimal target rearrangement invariant space for potential type operators is considered in [4] by using -capacities. The problem of the mapping properties of the Riesz potential in optimal couples of rearrangement invariant spaces is treated in [5–7]. The optimal embeddings of generalized Sobolev type spaces into rearrangement invariant spaces are characterized in several papers [5, 8–21]. The characterization of the continuous embedding of the generalized Bessel potential spaces into the generalized Hölder-Zygmund spaces , when is a weighted Lebesgue space, is given in [22]. The optimal embeddings of Calderón spaces into the generalized Hölder-Zygmund spaces are characterized in [23].

The plan of the paper is as follows. In Section 2 we provide some basic definitions and known results. In Section 3 we characterize the embedding . The optimal quasinorms are constructed in Section 4.

2. Preliminaries

We use the notations or for nonnegative functions or functionals to mean that the quotient is bounded; also, means that and . We say that is equivalent to if .

Let be a quasinormed rearrangement invariant space as in the Introduction. There is an equivalent quasinorm that satisfies the triangle inequality for some that depends only on the space (see [24]). We say that the quasinorm satisfies Minkowski’s inequality if for the equivalent quasinorm ,
Usually we apply this inequality for functions with some kind of monotonicity.

Recall the definition of the lower and upper Boyd indices and . Let if and if , where , and let
be the dilation function generated by . Suppose that it is finite. Then
The function is submultiplicative, increasing, ; hence . We suppose that .

If we have by using Minkowski’s inequality that . In particular, if . For example, consider the Gamma spaces , , —positive weight, that is, a positive function from , with a quasinorm , where
Then . If , we write as usual instead of . Consider also the classical Lorentz spaces , ; if , . We suppose that .

Note that if , where is slowly varying, then . Recall that is slowly varying if for all the function is equivalent to an increasing function and the function is equivalent to a decreasing function.

In order to introduce the Hölder-Zygmund class of spaces, we denote the modulus of continuity of order by
where are the usual iterated differences of . When we simply write .

Let be a quasinormed space of locally integrable functions on the interval with the Lebesgue measure, continuously embedded in and , where is a monotone quasinorm on which satisfies Minkowski’s inequality. The dilation function generated by is given by
where
The choice of the space is motivated by the fact that is equivalent to a function . The function is submultiplicative, increasing and is decreasing and . Therefore the Boyd indices of are well defined
and they satisfy . In what follows, we suppose that .

For example, let . Here and is a slowly varying function, and , or simply if , is the weighted Lebesgue space with a quasinorm , where is given by (5). It turns out that .

Definition 1. Let and let stand for the space of all functions , defined on , that have bounded and uniformly continuous derivatives up to the order , normed by , where . (i)If for or for , then is formed by all functions in having a finite quasinorm
(ii)If , then consists of all functions in having a finite quasinorm
Here , is the characteristic function of the interval .

In particular, if , then coincides with the usual Hölder-Zygmund space (see [1]). Also, if , then .

Note that if , then is a -interpolation space for the couple , namely, , where . In particular, . By , we denote the characteristic function of the interval .

Recall some basic definitions from the theory of interpolation spaces [21]. Let be a couple of two quasinormed spaces, such that both are continuously embedded in some quasinormed space and let
be the -functional of Peetre. By definition, the -interpolation space has a quasinorm , where is a quasinormed function space with a monotone quasinorm on with the Lebesgue measure and such that . Then . If
we write instead of . Also, if then we write . By definition,

Theorem 3 (lifting principle). Let and let . Then

Proof. Let . Since
it follows
Hence
To prove the reverse, we use the formula (see [25, page 342])
Then applying Minkowski’s inequality and , we get
Since , we derive

Remark 4. The relation (19) is always true. But if then the reverse might not be true. For example, let , , , and if and if . Then , but .

It will be convenient to introduce the classes of the domain and target quasinorms, where the optimality is investigated. Let consist of all domain quasinorms that are monotone, satisfying Minkowski’s inequality, , if and the condition (30) below for , and for . Let consist of all target quasinorms that are monotone, satisfy Minkowski’s inequality, , if and

We use the following definitions.

Definition 5 (admissible couple). We say that the couple is admissible if when , and if for . Moreover, is called domain quasinorm (domain space), and is called target quasinorm (target space).

Definition 6 (optimal target quasi-norm). Given the domain quasinorm , the optimal target quasi-norm, denoted by , is the strongest target quasi-norm; that is,
for any target quasinorm such that the couple , is admissible. Since , we call the optimal Hölder-Zygmund space.

Definition 7 (optimal domain quasi-norm). Given the target quasinorm , the optimal domain quasi-norm, denoted by , is the weakest domain quasi-norm; that is,
for any domain quasinorm such that the couple , is admissible.

Definition 8 (optimal couple). The admissible couple , is said to be optimal if both and are optimal.

3. Admissible Couples

Here we give a characterization of all admissible couples , . We start with the main estimate. For , see also [26].

Theorem 9. Let and . Then
where .

Proof. We use the embedding
whence
Then (26) follows from the basic formula [25, page 360]
and the reiteration formula of Holmstedt for the -functional [25, page 310].

Now we discuss the embedding . For more general results are proved in [27, Chapter 4].

Theorem 10. A necessary and sufficient condition for the embedding , is the following one
where

Proof. The conditions (30) and (26), (27) imply the embedding and if . On the other hand, by Marchaud’s inequality (see [25], Theorem ), we have
It is easy to see that . Thus .Before proving the reverse, note that (30) is always satisfied if . Since
we have
Hence for ,
It remains to prove that if , , then (30) is true for . To this end we choose a test function as follows:
where and is in such that if and if . Then for , whence for for appropriate and , whence ; therefore . This implies
Analogously, since , , we have
Also . Thus (30) is proved.

Remark 11. Similar arguments show that , if and only if .

Theorem 12. The couple , is admissible if and only if
where

Proof. Step 1 (sufficiency of (39)). If then it is clear that the embedding follows from (39), (26), and (27). Let now , . Then (39) for implies . Hence for .Step 2 (necessity of (39) when ). Now we prove that the embedding implies (39) for . To this end we choose the test function as in (36).Let . We split , , , . First we prove that for some large ,
Indeed, we have and for if is large enough. Hence (41) follows. Further,
Since
and for , we get
Therefore
To solve the integral inequality (45) for , we set and rewrite it as . If , then we get the differential inequality . If , then , whence . Therefore
Hence by using Minkowski’s inequality and choosing large enough, we obtain
On the other hand, from (46), it follows that
Hence, using also (30), we get
Thus, if is given, then (50), (38) imply (39).Step 3 (necessity of (39) when , ). Now we prove that the embedding , , implies (39) for . To this end we choose the test function in the form
where and is the same as in (36). Note that . Then as before we get (37), and
for . Hence
On the other hand, using the arguments from Step 2 but for the function , , we obtain
Thus, if , is given, then (54), (53) imply (39) for .

Theorem 13. Let . Then the couple , is admissible for if and only if
Moreover, (55) is equivalent to

Proof. Let be an admissible couple for . Then . Since
we have, by applying (39) for ,
Using also (16) and , we obtain . Further, as in the proof of the previous theorem, this embedding implies (56). Finally,
Indeed, and
Applying Minkowski’s inequality, we get, since ,
For the reverse, we notice that . Then

4. Optimal Quasinorms

Here we give a characterization of the optimal domain and optimal target quasinorms.

4.1. Optimal Domain Quasinorms

Definition 14 (construction of an optimal domain quasi-norm). For a given target quasinorm , we set

Note that and if . Hence .

Theorem 15. The quasinorm belongs to , the couple , is admissible, and the domain quasinorm is optimal. Moreover, the target quasinorm is also optimal and

Proof. It is easy to check that . Further, the couple , is admissible since , . Moreover, is optimal, since for any admissible couple , we have , where . Therefore for ,
To prove that is also optimal, let be an arbitrary admissible couple. Then
It is enough to check that
Let . (The case is easier.) We introduce a better function . Then is quasiconcave; therefore and . By changing the variables, we get with . Then
Thus (67) is proved. To prove the equivalence (64), we use and Minkowski’s inequality as follows:
whence .

Remark 16. Let and . Then the couple , is optimal.

Example 17. Consider the space , where and . Using Theorem 15, we can construct an optimal domain , where
and . Hence , and this couple is optimal. Also if is slowly varying. Note that if , then .

Example 18. Let , where and and let
Then by Theorem 15, the domain is optimal and the couple is optimal. In particular, the couple is optimal. If , , this means that the embedding is optimal.

Example 19. Let be as in the previous example. Since
it follows that the couple is admissible. In order to prove that is optimal, take any , and define from . Then and . On the other hand
Hence ; therefore is optimal.

Example 20 (case ). Let , , where and and let . Using Remark 16, we can construct an optimal domain and this couple is optimal. Also if is slowly varying.

4.2. Optimal Target Quasinorms

Definition 21 (construction of the optimal target quasi-norm). For a given domain quasinorm , we set
Note that .

Theorem 22. The target quasinorm belongs to , the couple is admissible, and the target quasinorm is optimal.

Proof. The property “ implies ” follows from (30). Also, since it is easy to check that . The couple is admissible since , . Suppose that the couple , is admissible. Then , . Therefore if , , then , whence , . Hence is optimal.

Theorem 23 (supercritical case). If and , then
Moreover, the couple is optimal.

Proof. If , , then, by Minkowski’s inequality and since , it follows
Hence, taking the infimum, we get .On the other hand, for , we have , . Since it follows .The domain quasinorm is also optimal since for ,

Example 24. Consider the space , , . Then by Theorem 23 the couple , is optimal. In particular, using also Theorem 13, the embedding , , , is optimal.

In the critical case we do not know how to simplify the optimal target quasi-norm, defined in (74). Instead, we can construct a large class of domain quasinorms and the corresponding optimal target quasinorms by using extrapolation from the super-critical case. Recall some basic definitions and results from the extrapolation theory [3]. Let be a couple of quasi-Banach spaces. The sigma extrapolation space , -positive weight, , , -positive decreasing weight, consists of all such that , , , with a quasinorm
where the infimum is taken with respect to all representations .

This space can be characterized as an interpolation space.

Theorem 25 (see [3]). Let , -slowly varying, . Then
where
and if , if .

Our main result is the following one.

Theorem 26. Let , , -slowly varying weight, , , , . We suppose that and . Then this couple is admissible and the target quasinorm is optimal.

Proof. Step 1 (admissibility). Since , it will be enough to check that
where
Applying Minkowski’s inequality we obtain for , -slowly varying weight,
In order to extrapolate these inequalities, we write
This is true since
Let and (convergence in ), where . Then , whence and for , we have
where . We can write
and using also Hölder’s inequality if , we get
where
Hence
Since
we get
whence
where is given by (80) with and . It is easy to calculate these weights, see [3]. We have
Then for we get
Hence (81) is proved.

Step 2 (optimality of the target quasi-norm). We want to prove that is an optimal target quasi-norm. It is sufficient to see that
where is defined by (74). To this end for any such we construct an such that and . Let . (The case is analogous, see Example 19.) Let . Then and . On the other hand, for ,
since